MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ofmresval Structured version   Unicode version

Theorem ofmresval 6534
Description: Value of a restriction of the function operation map. (Contributed by NM, 20-Oct-2014.)
Hypotheses
Ref Expression
ofmresval.f  |-  ( ph  ->  F  e.  A )
ofmresval.g  |-  ( ph  ->  G  e.  B )
Assertion
Ref Expression
ofmresval  |-  ( ph  ->  ( F (  oF R  |`  ( A  X.  B ) ) G )  =  ( F  oF R G ) )

Proof of Theorem ofmresval
StepHypRef Expression
1 ofmresval.f . 2  |-  ( ph  ->  F  e.  A )
2 ofmresval.g . 2  |-  ( ph  ->  G  e.  B )
3 ovres 6424 . 2  |-  ( ( F  e.  A  /\  G  e.  B )  ->  ( F (  oF R  |`  ( A  X.  B ) ) G )  =  ( F  oF R G ) )
41, 2, 3syl2anc 661 1  |-  ( ph  ->  ( F (  oF R  |`  ( A  X.  B ) ) G )  =  ( F  oF R G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    X. cxp 4997    |` cres 5001  (class class class)co 6282    oFcof 6520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-xp 5005  df-res 5011  df-iota 5549  df-fv 5594  df-ov 6285
This theorem is referenced by:  psradd  17803  dchrmul  23248  ldualvadd  33926
  Copyright terms: Public domain W3C validator